Linear Operators: General theory |
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Page 373
... Riesz [ 2 ; p . 456 ] . Lemma 4.2 is due to F. Riesz [ 8 ; p . 36 ] and was also used by Sz . - Nagy [ 5 ] . A similar argument may be applied to obtain this result in any uniformly convex B - space . This generalizes and abstracts a ...
... Riesz [ 2 ; p . 456 ] . Lemma 4.2 is due to F. Riesz [ 8 ; p . 36 ] and was also used by Sz . - Nagy [ 5 ] . A similar argument may be applied to obtain this result in any uniformly convex B - space . This generalizes and abstracts a ...
Page 387
... Riesz [ 9 ] . A detailed proof was given by Fréchet [ 5 ; III . p . 441 ] . The theorem for L „ [ 0 , 1 ] , 1 < p < ∞ , was demonstrated by F. Riesz [ 2 ; p . 475 ] . In the case of a finite measure space the theorem was established by ...
... Riesz [ 9 ] . A detailed proof was given by Fréchet [ 5 ; III . p . 441 ] . The theorem for L „ [ 0 , 1 ] , 1 < p < ∞ , was demonstrated by F. Riesz [ 2 ; p . 475 ] . In the case of a finite measure space the theorem was established by ...
Page 388
... Riesz [ 13 ] . THEOREM . If 1 < p < ∞ then a sequence { f } converges strongly to f in L , ( S , E , μ ) if and only if it converges weakly and in → f . This theorem remains valid in any uniformly convex B - space . Theorem 8.15 was ...
... Riesz [ 13 ] . THEOREM . If 1 < p < ∞ then a sequence { f } converges strongly to f in L , ( S , E , μ ) if and only if it converges weakly and in → f . This theorem remains valid in any uniformly convex B - space . Theorem 8.15 was ...
Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element ergodic exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ